Question: Simplify; express your answer in exponential form. Assume $z\neq 0, a\neq 0$. $\dfrac{{(z^{-3}a^{-3})^{5}}}{{z^{-2}a^{-2}}}$
Explanation: To start, try simplifying the numerator and the denominator independently. In the numerator, we can use the distributive property of exponents. ${(z^{-3}a^{-3})^{5} = (z^{-3})^{5}(a^{-3})^{5}}$ On the left, we have ${z^{-3}}$ to the exponent ${5}$ . Now ${-3 \times 5 = -15}$ , so ${(z^{-3})^{5} = z^{-15}}$ Apply the ideas above to simplify the equation. $\dfrac{{(z^{-3}a^{-3})^{5}}}{{z^{-2}a^{-2}}} = \dfrac{{z^{-15}a^{-15}}}{{z^{-2}a^{-2}}}$ Break up the equation by variable and simplify. $\dfrac{{z^{-15}a^{-15}}}{{z^{-2}a^{-2}}} = \dfrac{{z^{-15}}}{{z^{-2}}} \cdot \dfrac{{a^{-15}}}{{a^{-2}}} = z^{{-15} - {(-2)}} \cdot a^{{-15} - {(-2)}} = z^{-13}a^{-13}$